Bayes flips "effect given cause" into "cause given effect"
Bayes' theorem, named for Thomas Bayes and refined with Pierre-Simon Laplace, turns a familiar conditional probability around. Knowing how often a positive test appears when disease is present, plus how common the disease is, yields the chance the patient is truly ill given that positive result.
In probability, Bayes' theorem (also Bayes' law or rule) inverts conditionals: from the likelihood of evidence under a cause, recover the probability of the cause given the evidence. Bayesian inference uses the same flip to move from a model's likelihood to a posterior over model settings. The algebraic core equates two expansions of a joint probability, yielding P(A|B) = P(B|A)P(A)/P(B) when P(B) ≠ 0. With mutually exclusive, exhaustive alternatives Ai, the law of total probability expands the denominator into a weighted sum of likelihoods.
Thomas Bayes, a minister and mathematician, framed an algorithm (his Proposition 9) for bounding an unknown binomial parameter from evidence. After his death Richard Price edited the manuscript for two years; it was read at the Royal Society on 23 December 1763 and printed in Philosophical Transactions under a long eighteenth-century title about solving a problem in the doctrine of chances. Price's introduction sketched philosophical ground for Bayesian statistics; in 1765 he became an FRS, and a 27 April letter to Benjamin Franklin applied the ideas to population and life annuities. Laplace independently rebuilt and extended the update from prior to posterior in 1774 and in his 1812 Théorie analytique des probabilités, largely shaping the Bayesian reading of probability. Harold Jeffreys later axiomatized the framework, writing in 1973 that Bayes' theorem is to probability what Pythagoras is to geometry.
Priority debates continue: Stephen Stigler floated Nicholas Saunderson as an earlier discoverer—a claim F. Thomas Bruss disputed while still defending the Bayes name. Martyn Hooper and Sharon McGrayne urge "Bayes–Price" credit for Price's discovery, correction and use of the essay. Charles Sanders Peirce (1878) logged the logarithm of the odds factor as "weight of evidence"; Alan Turing in the early 1940s called a related factor the "factor in favour of a proposition."
A clinical sketch shows the stakes. Let E be disease and F a positive test; posterior disease risk mixes prevalence, sensitivity and false-positive behaviour. If pancreatic-cancer incidence is 1/100000 while 10 of 99999 healthy people share the same symptoms, having those symptoms raises the chance of that cancer only to about 9.1%—a sharp reminder that a high true-positive rate does not make a rare disease likely.
Source: Bayes' theorem
